1

Blind Calibration of a Bandpass SamplingNonuniformly Interleaved Two-Channel ADC

Bernard C. Levy, Fellow, IEEE,Anthony Van Selow and Mansoor S. Wahab,Student Member, IEEE,

Abstract—A method was proposed in [1] to sample directly thecomplex envelope of a bandpass signal by using two nonuniformlyinterleaved analog-to-digital converters (ADCs). The computationof the envelope samples requires two digital FIR filters and adiscrete-time demodulator. To mitigate the effect of gain timingmismatches, we describe a blind calibration technique whichassumes that there exists a frequency band where the envelope sig-nal has no power, due for example to oversampling. Mismatchesgive rise to errors in this band, which are extracted and usedto estimate adaptively the gain and timing skew mismatches.Computer simulations for an envelope consisting of a sum ofcomplex exponentials and for a bandlimited white noise envelopedemonstrate the effectiveness of the proposed calibration method.

Keywords—adaptive filtering, bandpass sampling, blind calibra-tion, direct sampling, software defined radio, stochastic gradient,time interleaved A/D converter

I. I NTRODUCTION

Due to their lower hardware complexity, direct bandpasssampling front ends have become attractive for softwaredefined radio and radar, and for global navigation satellitesystem (GNSS) receivers. However, the implementation offlexible high-resolution bandpass sampling systems presentssome challenges. If a single ADC is used, andB representsthe occupied bandwidth of the signal of interest (which differsfrom its maximum frequency), alias-free reconstruction ofthebandpass signal is not guaranteed for all sampling frequenciesΩs above the Nyquist frequency2B [2], [3], [4, Sec. 6.4].In addition to being greater than2B, Ωs needs to satisfyconditions which ensure that aliasing does not take placebetween the negative and positive frequencies of the bandpasssignal to be sampled. It was shown in [2], that this difficultycan be overcome by using second-order sampling, i.e. time-interleaved sampling, where two separate ADCs operating witha time skew sample the signal with frequencyΩ′

s = Ωs/2. Inthis case, except for certain forbidden values of the timingoffset between the two ADCs, the bandpass signal can bereconstructed from the two time-interleaved sample sequencesfor all sampling frequenciesΩs above2B.

Whereas [2] and subsequent works [5], [6] focus on thereconstruction of the bandpass signal signal itself from itssamples, for modulated signal, it is often of greater interest

This work was supported by NSF Grant ECCS-1444086.Bernard C. Levy and and Anthony Van Selow are with the Department

of Electrical and Computer Engineering, 1 Shields Avenue, University ofCalifornia, Davis, CA 95616 (emails: bclevy, apvanselow@ucdavis.edu). Afterreceiving a BS at UC Davis, Mansoor Wahab is now a graduate student inthe Department of Electrical and Computer Engineering at Cornell University,Ithaca, NY 14853 (email: sw798@cornell.edu).

to obtain the sampledI and Q signal components, i.e., thesampled complex envelope of the signal. A method is describedin [1] to compute the sampled complex envelope of a bandpasssignal directly from the sequences produced by a two-channeltime-interleaved ADC (TIADC) with timing offsetdT ′

s, where0 < d < 1 and T ′

s = 2π/Ω′s denotes the sub-ADC sampling

period. No assumption is made about the carrier frequencyΩc,signal bandwidthB, sampling frequencyΩs and timing offsetd, except that the sampling frequencyΩs should be above2B and the carrier frequencyΩc > B/2, which ensures thatthe signal considered is a bandpass signal. However, certaintiming offests are forbidden, including the offsetd = 1/2corresponding to uniform interleaving, since this case theinterleaved ADC is functionally equivalent to a single ADC,to which the limitations noted in [3] are applicable.

The computational technique described in [1] for evaluatingthe complex envelope samples from the sub-ADC samples re-lies on two real digital FIR filter and a digital demodulator.Thefilters need to be recomputed if the frequency band of interestchanges, but since the filters are obtained by windowing knownclosed form impulse responses, the recomputation is rathersimple, making the proposed technique well suited for softwaredefined radio or radar receivers. However, the performancecomplex envelope computation algorithm is affected by timingoffset mismatches, i.e. differences between the actual offsetand the one used to compute the FIR filters. The sourceof this sensitivity is that the model of a bandpass samplingnonuniformly interleaved ADC involves a2πℓd phase shift,where if k denotes the Nyquist zone of the carrier frequencyΩc, ℓ = ⌊k/2⌋. Hence whenℓ is large, small timing offsetmismatches give rise to large phase errors which significantlyaffect the proposed envelope computation technique.

Another reason for examining the effect of timing mis-matches is that since the uniform timing skewd = 1/2is among the forbidden timing offset values, the sub-ADCscannot share the same sample and hold (S/H). The use ofdifferent S/Hs in the two sampling paths makes it difficult toachieve exactly the desired timing offsetd. It is therefore im-portant to develop a timing and gain offset calibration schemeadapted to the bandpass sampling nonuniformly interleavedADC architecture considered. Calibration techniques can dedivided in two categories, depending on whether the normalADC operation is is suspended during calibration phases.Foreground calibration techniques interrupt the ADC operationby using a known test signal and identify mismatches byusing equalizer type of techniques [7], [8]. Background orblind methods identify mismatches while the ADC is operatingnormally. They rely on some a priori signal knowledge, suchas the absence of signal energy in certain frequency bands

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(this can be achieved for example if the signal is oversampled[9], [10]) or the wide-sense stationarity of the sampled signal[11]. Foreground methods have the disadvantage of loweringthe ADC throughput, whereas background methods have theadvantage of being able to track continuously ADC parameterchanges, but they require additional digital filtering operations.

Over the last 10 years a number of blind calibration tech-niques have been proposed for calibrating gain and timingmismatches [9], [11]–[14] or bandwidth mismatches [10],[15]–[17] in baseband sampling uniformly interleaved ADCs.In contrast, little attention has focused on bandpass sam-pling ADCs, except for [18]–[20] which propose a semi-blind scheme where an analog test signal is injected in thesignal prior to digitization, and then removed from the TIADCoutput. This approach requires oversampling and making surethat the aliased images of the injected signal do not overlapthe spectrum of the signal to be sampled. More recently, ablind method described in [21] extends the pseudo-aliasingcalibration approach proposed in [14] to the bandpass case.However, all the results obtained in the bandpass case thusfar concern uniformly interleaved ADCs where the goal is tosample the bandpass signal itself instead of its envelope. Wepropose here a different blind calibration method tailoredtothe envelope sampling technique of [1] which leverages thedigital filtering operations used to compute the envelope inorder to perform the TIADC calibration. As in [9], [10] it isassumed that the signal is slightly oversampled, by say 20%.This creates a frequency band where the complex envelopedoes not have any spectral content. Therefore the presence ofenergy in this band in the reconstructed envelope indicatesthat the TIADC gain and timing skew estimates used inthe reconstruction are incorrect. By implementing a complexbandpass filter which extracts the signal power in this band,weconstruct an error signal whose power is minimized adaptivelyto estimate the gain and timing offset mismatches.

The paper is organized as follows. The complex envelopesampling scheme of [1] is reviewed in Section II. The blindcalibration approach we propose is described in Section III.Simulation results are presented in section IV for an envelopeformed by the sum of complex tones and for a bandlimitedwhite noise envelope. It is shown that calibration leads to asignificant improvement in both the SNDR and MSE of thecomplex envelope sampling method. Finally, conclusions arepresented in Section V.

II. COMPLEX ENVELOPE SAMPLING

Consider a bandpass signal

xc(t) = ac(t) cos(Ωct) − bc(t) sin(Ωct)

= ℜ[cc(t)ejΩct] = |cc(t)| cos(Ωct + ∠cc(t)) , (1)

wherecc(t) = ac(t) + jbc(t) (2)

denotes the complex envelope ofxc(t) and the in-phase andquadrature componentsac(t) and bc(t) are baseband signalswith bandwidthB/2. If the carrier frequencyΩc > B/2 (it isin general much larger), the Fourier spectrumXc(jΩ) of xc(t)

is contained in two disjoint positive and negative frequencybands[ΩL,ΩH ] and [−ΩH ,−ΩL] with ΩL = Ωc − B/2 andΩH = Ωc + B/2, so that the occupied bandwidth ofxc(t) isΩH −ΩL = B. To sample the complex envelope, we considerthe two-channel time-interleaved ADCs shown in Fig. 1.

ADC1

ADC2

xc(t)

x1(n)

x2(n)

nT ′s

(n − d)T ′s

Fig. 1. Time-interleaved sampling of bandpass signalxc(t).

T ′s denotes the sampling period of the sub-ADCs, which

have therefore sampling frequencyΩ′s = 2π/T ′

s = Ωs/2,whereΩs denotes the sampling frequency of the overall ADC.It is assumed thatΩs > 2B, or equivalently Ω′

s > B.The timing offset between the two ADCs isD = dT ′

s with0 < d < 1. Ford = 1/2, the combination of the two sub-ADCsforms a uniform ADC with sampling periodTs = T ′

s/2, butfor d 6= 1/2, the overall ADC has a nonuniform but periodicsampling pattern. Although the two sub-ADCs are assumed tobe matched, due to manufacturing imperfections, the secondsub-ADC has a relative gaing compared to the first.

Let

ℓ = round(Ωc

Ω′s

)

. (3)

so thatΩc belongs to the frequency band[(ℓ − 1/2)Ω′s, (ℓ +

1/2)Ω′s]. Since this frequency band corresponds to the location

of the ℓ-th image copy of a sampled baseband signal, it isreferred to here as theℓ-th image band. In the following itis assumed thatℓ ≥ 1, so that the signalxc(t) is a bandpasssignal. Since the band[(ℓ− 1/2)Ω′

s, (ℓ+1/2)Ω′s] corresponds

to the 2ℓ-th and 2ℓ + 1-th Nyquist zones of the sub-ADCs,ℓ can be expressed in terms the indexk of the Nyquist zonewhereΩc is located asℓ = ⌊k/2⌋. If we consider the discretemodulation frequency

ωb = ΩcT′

s mod 2π =(Ωc

Ω′s

− ℓ)

2π , (4)

so that−π < ωb ≤ π, and if c(n) = cc(nT ′s) denotes the

sampled complex envelope, it is shown in [1] that the sampledsequencesx1(n) and x2(n) produced by the two sub-ADCsare related toc(n) by the model shown in Fig. 2. In this model,the frequency response

F (ejω) = ge−jωd (5)

with −π < ω ≤ π represents the relative gain and timingskew between the two-sub-ADCs, andG(ejω) is a piecewise

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constant phase shift. Forωb > 0, it can be expressed as

G(ejω) =

e−j2π(ℓ+1)d −π ≤ ω < −π + ωb

e−j2πℓd −π + ωb ≤ ω < π ,

and forωb < 0, we have

G(ejω) =

e−j2πℓd −π ≤ ω ≤ π + ωb

e−j2π(ℓ−1)d π + ωb < ω < π .

Theℜ. operation in Fig. 2 takes the real part of a complexsignal. Thus, from a software defined radio perspective,G(ejω)depends on the carrier frequencyΩc of the signal of interest,since it depends on its image zone indexℓ, and on its relativelocation ωb within this band, butF (ejω) remains the samesince it depends only on the sub-ADCs relative characteristics.

G(ejω) ℜ· F (ejω)

ℜ·

ejωbn

c(n) x1(n)

x2(n)

Fig. 2. Discrete-time bandpass sampling model

Let

θ =

[

gd

]

denote the vector formed by the sub-ADC parameters. It wasshown in [1] that the complex envelopec(n) can be evaluatedby passing the sub-ADC sequencesx1(n) andx2(n) throughdigital digital filtersH1(e

jω, d) andH2(ejω,θ) whose outputs

are summed and then digitally demodulated withe−jωbn asshown in Fig. 3. The filterH1(e

jω, d) depends ond only andis given by

H1(ejω, d) = 1 − j cot(2πℓd) (6)

for |ω| ≤ π − |ωb| and

H1(ejω, d) = 1 − j cot(π(2ℓ + sgn(ωb))d) (7)

for π− |ωb| < |ω| ≤ π. Similarly H2(ejω, θ) depends on both

g andd and takes the form

H2(ejω,θ) = j

ejωd

g sin(2πℓd)(8)

for |ω| ≤ π − |ωb| and

H2(ejω,θ) = j

ej(ω−πsgn(ω))d

g sin(π(2ℓ + sgn(ωb)))(9)

for π − |ωb| < |ω| ≤ π. The IIR impulse responses of filtersH1(e

jω, d) andH2(ejω,θ) are computed in [1] and are given

respectively by

ℜh1(n, d) = δ(n)

ℑh1(n, d) = − cot(π(2ℓ + sgn(ωb))d)δ(n)

(cot(π(2ℓ + sgn(ωb))d) − cot(2πℓd))

×sin((π − |ωb|)n)

πn, (10)

andℜh2(n,θ) = 0

ℑh2(n,θ) =sin((π − |ωb|)(n + d))

gπ sin(2πℓd)(n + d)

−sin((π − |ωb|)(n + d) − πd)

gπ sin(π(2ℓ + sgn(ωb))d)(n + d). (11)

H1(ejω, d)

H2(ejω,θ)

x1(n)

x2(n)e−jωbn

c(n)r(n,θ)

Fig. 3. Recovery ofc(n) from x1(n) and x2(n) by filtering and digitaldemodulation.

III. B LIND TIADC CALIBRATION METHOD

In practice, clock jitter and semiconductor process imperfec-tions make it impossible to match exactly the two sub-ADCsto their design specifications. For a desired sub-ADC timingskew ofd0, due to circuit imperfections, the actual timing skewis d and the sub-ADC relative gaing differs from its from itsideal valueg0 = 1. Typically, the mismatches

γ= g − 1 , δ

= d − d0 (12)

are small, say about 1% or less, but they contribute to asignificant loss of ADC resolution. Accordingly, it is desirableto implement calibration algorithms to estimateg and d. Inthe following the estimated parameter vector at a given timeis denoted as

θ =

[

g

d

]

,

where g and d denote the gain and timing offset estimates.Since mismatches are small, it is assumed throughout theremainder of this paper thatθ, θ, and the nominal parametervector

θ0 =

[

1d0

]

are close to each other. The calibration method we considerwill match the gains of the two sub-ADCs, to ensure thatgis properly compensated, instead of attempting to correct themismatch of each channel gain with respect to its nominalvalue. This choice of calibration strategy is dictated by our

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decision to adopt a blind calibration approach since, in theabsence of an external reference signal, it is impossible todetect gain errors which are common to both channels.

Since the exact gaing and timing skewd are unknown,the estimated valuesg and d can be substituted for theexact parameter values in reconstruction network of Fig. 3used to evaluate the complex envelope samples. However toavoid updating all the coefficients of filtersH1(e

jω, d) andH2(e

jω, θ) each time the parameter vector vectorθ is updated,we exploit the fact that the estimated mismatches

γ = g − 1 , δ = d − d0

are small by performing first order expansions

H1(ejω, d) = H10(e

jω) + δH11(ejω) (13)

H2(ejω, θ) = (1 − γ)[H20(e

jω) + δH21(ejω)] (14)

with

H10(ejω) = H1(e

jω, d0) , H20(ejω) = H2(e

jω,θ0)

H11(ejω) =

∂H1

∂d(ejω, d0) , H21(e

jω) =∂H2

∂d(ejω,θ0)

of the frequency responses and impulse responses ofH1 andH2 in the vicinity of the nominal parameter values. Theimpulse responses of gradient filtersH11(e

jω) and H21(ejω)

are evaluated in Appendix A. The multiplicative term1 − γin (14) is due to the fact thatH2(e

jω, θ) is proportional tog−1, so that a local expansion in the vicinity ofg0 = 1results in a1 − γ multiplicative factor. Although first orderterms are sufficient for small to moderate values ofℓ, ifmore accurate approximations are needed second order termscould be retained. The resulting adaptive implementation ofthe reconstruction filters is shown in Fig. 4. Note that the fourfilters Hij(e

jω) with i = 1, 2 andj = 0, 1 are fixed and onlythree taps equal to estimated mismatchesδ and γ need to beadapted.

Since the estimated parameter vectorθ is used in the recon-struction network of Fig. 4, the output signalr(n, θ) obtainedafter summing the outputs of filtersH1(e

jω, d) andH2(ejω, θ)

differs from the exact signalr(n,θ) = c(n)ejωbn that wouldbe obtained for the correct parameter vector. Therefore wecan use differences betweenr(n, θ) and the correct signalr(n,θ) to calibrate the TIADC. The blind calibration schemewe propose estimatesθ adaptively by extracting fromr(n, θ)an error signale(n, θ) whose power provides a measure ofhow far the estimated vectorθ is from the true vectorθ. It isassumed that the complex envelopecc(t) is oversampled withoversampling ratio

α =Ωs − 2B

Ωs

. (15)

In this case the periodic discrete-time Fourier transform(DTFT) C(ejω) of the sampled envelope sequencec(n) satis-fies

C(ejω) = 0

over interval[(1−α)π, (1+α)π] and all its2π-shifted copies.This implies that when the estimated parameter vectorθ coin-cides with the correct vectorθ, the signalr(n,θ) = c(n)ejωbn

has no frequency content in the bandI = [ωb +(1−α)π, ωb +(1 + α)π] and its 2π-shifted copies. Consider therefore thediscrete-time bandpass filter

HBP(ejω) =

1 ω ∈ I mod (2π)0 otherwise . (16)

It can be viewed as an ideal lowpass filter for the band[−απ, απ] modulated byωb + π, so that its IIR impulseresponse is given by

hBP (n) = (−ejωb)n sin(απn)

πn

Then if ∗ denotes the convolution operation, let

e(n, θ) = hBP (n) ∗ r(n, θ) (17)

denote the error signal obtained by extracting fromr(n, θ)its frequency content in the bandI. Since e(n, θ) = 0wheneverθ = θ, the estimation scheme we propose minimizesadaptively the power

J(θ) =1

2E[|e(n, θ)|2] (18)

of complex signale(n, θ). This approach is justified rigorouslyin Appendix B by showing that if the complex enveloperandom processc(n) is wide-sense stationary (WSS), in thevicinity of θ J(θ) can be approximated by a positive definitequadratic function ofγ − γ and δ − δ, providedc(n) satisfiescertain spectral frequency content conditions. This ensures thatby minimizingJ(θ) iteratively, starting with the nominal value

θ(0) =

[

1d0

]

, (19)

the iterates will converge to the correct true parameter vectorθ.

If the gradient of functionJ(θ) is available, which istypically not the case since the evaluation of the ensembleaverage in (18) requires knowledge of the statistics of theenvelope processc(n), J(θ) can be minimized by using asteepest descent iteration of the form

θ(n + 1) = θ(n) − µ∇θJ(θ(n)) (20)

whereµ > 0 denotes the iteration step size, and

∇θJ(θ) =

[

∂∂g

∂

∂d

]

J(θ) (21)

represents the gradient ofJ(θ). In the absence of the gradientof J(θ), we apply astochastic gradient approximation[22]–[24], where the expectation in (18) is replaced by its instanta-neous value, which yields the adaptive estimation algorithm

θ(n + 1) = θ(n) − µℜe∗(n, θ(n))∇θe(n, θ(n)) (22)

5

H11(ejω)

c(n)

H10(ejω)

H20(ejω)

H21(ejω)

δ

δ

HBP(ejω)e11(n)

HBP(ejω)e21(n)

HBP(ejω)e2(n, θ)

HBP(ejω)e(n, θ)

r(n, θ)

e−jωbn

x1(n)

x2(n)

1 − γ

Fig. 4. Adaptive reconstruction filter implementation and error signal computation.

with initial condition (19). Typically the value ofµ used forthis recursion is much smaller than for the steepest descentalgorithm (20), and since the recursion (22) can be interpretedas computing the average of gradient∇

θ|e2(n, θ)|2/2, the

approximation used to go from (20) to (22) just consists inreplacing an ensemble average by a finite time average.

The recursion (22) requires the computation of the gradientof e(n, θ) with respect toθ. To perform this evaluation, it isconvenient to decompose the error signal as

e(n, θ) = e1(n, d) + e2(n, θ) (23)

with

e1(n, d) = k1(n, d) ∗ x1(n)

e2(n, θ) = k2(n, θ) ∗ x2(n) , (24)

wherek1(n, d) and k2(n, θ) denote respectively the impulseresponses of filters

K1(ejω, d) = HBP(ejω)H1(e

jω, d)

≈ HBP(ejω)[H10(ejω) + δH11(e

jω)] . (25)

and

K2(ejω, θ) = HBP(ejω)H2(e

jω, θ)

≈ (1 − γ)HBP(ejω)[H20(ejω) + δH21(e

jω)] . (26)

By using the first order expansions (25), we find that the errorcomponent signale1(n, d) can be represented as as

e1(n, d) = e10(n) + e11(n)δ , (27)

wheree1j(n) is the signal obtained by passingx1(n) throughfilter HBP(ejω)H1j(e

jω) with j = 0, 1. Similarly, by using(26), e2(n, θ) can be expressed as

e2(n, θ) = (1 − γ)[e20(n) + e21(n)δ] , (28)

wheree2j(n) is the signal obtained by passingx2(n) throughfilter HBP(ejω)H2j(e

jω) for j = 0, 1.Then, by observing that onlye2(n, θ) depends ong, we find

that∂e

∂g(n, θ) = −

1

1 − γe2(n, θ) . (29)

Similarly, the derivative ofe(n, θ) with respect tod can be

6

expressed as

∂e

∂d(n, θ) = e11(n) + (1 − γ)e21(n) . (30)

The parameter vector estimateθ(n) is evaluated adaptively bysubstituting gradient expressions (29) and (30) inside recursion(22). Since the gradients require both the evaluation of theerrore(n, θ) and its componentse2(n, θ), e11(n) and e21(n), andsince the first-order expansions require the implementation offixed filtersHij for i = 1, 2 andj = 0, 1, eight digital filtersare ultimately required to implement the proposed envelopesampling and calibration algorithm, as shown in Fig. 4. Theideal impulse responses of filtersHij , i = 1, 2 j = 0, 1 andHBP are noncausal and IIR, but causal FIR implementationsof these filters can be obtained by windowing and inclusionof appropriate delays. It is also worth observing that theimplementation of the four copies of bandpass filterHBP usedto generate error componentse2, e11 and e21 and errore isrequired only during calibration phases. After convergence, ifthe values ofg andd are not expected to change, the estimationalgorithm (20) and associated bandpass filters can be turnedoff to save power, until a new calibration update is required.However, if power consumption is not a significant considera-tion, it may be beneficial to keep running the recursion (22) inthe background to track drifts in the values ofg andd due tovariations in TIADC operating conditions, such as temperaturechanges.

IV. SIMULATIONS

A. Sum of Complex Tones Complex Envelope

To illustrate the proposed TIADC calibration method, weconsider first a bandpass signal withFc = Ωc/(2π) =5.15GHz, whose continuous-time envelope

cc(t) = 2 cos(400 × 106 × 2πt)

+j[sin(400 × 106 × 2πt) + cos(175 × 106 × 2πt)]

=3

2ej400×106

×2πt +1

2e−j400×106

×2πt

+j

2[ej175×106

×2πt + e−j175×106×2πt] (31)

contains 4 complex tones and has bandwidthB/(4π) =400MHz. The sub-ADC sampling frequencyF ′

s = Ω′s/(2π) =

1GHz is aboveB/(2π), and the oversampling ratioα = 0.2.Since

Fc = 5F ′

s + 150 ,

we haveℓ = 5 and

ωb =150

1000× 2π = 0.3π .

The discrete-time envelope obtained by samplingcc(t) withsampling periodT ′

s = 1/F ′s is

c(n) = cc(nT ′

s) =3

2ej0.8πn +

1

2e−j0.8πn

+j

2[ej0.35πn + e−j0.35πn] (32)

This signal has four tones located at±0.8π and±0.35π, butthe tone at0.8π has an amplitude three times larger than thetones at−0.8π and±0.35π. Note that the tone located at0.35πis inside the frequency band

IT = [0.8π, 1.2π] − 0.6π = [0.2π, 0.6π]

defined in (53), wherec(n) is required to have some en-ergy to ensure that the proposed calibration method workssatisfactorily. Also, the passband of filterHBP(ejω) is I =[0.8π, 1.2π] + 0.3π mod (2π) = [−0.9π,−0.5π].

Although the nominal timing offset used in the TIADCreconstruction network of Fig. 4 isd0 = 0.425, timing and gainmismatchesδ = d−d0 = −0.25×10−2 andγ = g−1 = 10−2

are present. The two sub-ADC sequences are therefore givenby

x1(n) = ℜc(n)ejωbn + v1(n)

x2(n) = ℜ(f ∗ c)(n)ejωb(n−d) + v2(n) , (33)

wheref(n) denotes the impulse response ofF (ejω) and thezero mean white noisesv1(n) and v2(n) model the effect ofthermal and quantization noises. The sub-ADC SNR is61.8dB.

The reconstruction filtersHij(ejω) with i = 1, 2 and

j = 0, 1 appearing in Fig. 4 are obtained by windowing theIIR impulse responses of these filters with a Kaiser windowof length 61 and parameterβ = 6. The magnitude of thefrequency responses of the resulting FIR filters are shown inFig. 5. The small notches of these filters (except forH11) atthe frequencyπ−|ωb| = 0.7π are due to phase discontinuitiesof the reconstruction filters (except forH11) at this frequency.Note that since the magnitude of the gradient filtersHi1(e

jω)with i = 1, 2 is approximately proportional toℓ, for the valueℓ = 5, the gradient filters have a large magnitude, suggestingthat for large values ofℓ, higher terms should be retained inexpansions (13) and (14) of the reconstruction filters. To ensurethat the transition region of the bandpass filterHBP(ejω) doesnot extend outside[−0.9π,−0.5π], we select a valueα = 0.15(corresponding to a narrower passband of[−0.85π,−0.55π])in the impulse responsehBP(n) and employ a Kaiser windowof length81 and parameterβ = 8.

For the 4 tones complex envelope considered, the blindestimation algorithm (22) is applied to a sequence of lengthL = 5 × 104 samples. The initial mismatches are selected asγ = 0 and δ = 0, which corresponds to the selecting thenominal TIADC parameters. Instead of using the same stepsize for both the gain and timing offset mismatches, we selectµγ = 10−3 andµδ = 10−5 for improved speed of convergenceand estimation accurcay. This choice can be interpreted asreplacing the steepest descent algorithm by a quasi-Newtonalgorithm. The resulting gain and time-delay mismatches areshown in Fig. 6 and Fig. 7 respectively. Note that the finalestimated gain mismatch valueγ(L) = 0.77×10−2 underesti-mates the gain mismatch by more than 20%, but the estimatedtiming offset mismatchδ(L) = −0.27× 10−2 is very close tothe exact mismatch valueδ.

To demonstrate the performance improvement of the pro-posed calibration algorithm, Fig. 8 and Fig. 9 show the powerspectral densities (PSDs) of the reconstructed envelopec(n)

7

0 0.51

1.2

1.4

1.6

1.8

f

MagnitudeofH

10

0 0.51

1.2

1.4

1.6

1.8

f

MagnitudeofH

20

0 0.545

50

55

60

65

f

MagnitudeofH

11

0 0.510

20

30

40

50

f

MagnitudeofH

21

Fig. 5. Magnitudes of FIR approximation of reconstruction filtersHij withi = 1, 2 andj = 0, 1 for a Kaiser window of lengthM = 61 and parameterβ = 6.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

1

2

3

4

5

6

7

8x 10

−3

# of samples

γ

Fig. 6. Gain mismatch estimates for a 4 tone complex input sequence oflength L = 5 × 104 samples, and adaptation step sizesµγ = 10−3 amdµδ = 10−5.

obtained before and after calibration. Before calibration, thesampled envelope is computed by using the reconstructionfilters H10(e

jω) and H20(ejω) computed for the nominal

TIADC values, whereas the calibrated sequencec(n) usesthe last104 estimatesγ(n) and δ(n) produced by the blindestimation algorithm. Note that as indicated by Fig. 6 and Fig.7, the blind estimation algorithm converges quickly, so that the

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

−3

−2.5

−2

−1.5

−1

−0.5

0x 10

−3

# of samples

δ

Fig. 7. Timing offset mismatch estimates for a 4 tone complex inputsequenceof lengthL = 5 × 104 samples, and adaptation step sizesµγ = 10−3 amdµδ = 10−5.

mismatch estimates obtained during the final104 simulationsamples fluctuate around the final values.

−0.5 0 0.5−140

−120

−100

−80

−60

−40

−20

0

20

f

PSD

ofc(n)(dB)

Fig. 8. PSD of the estimated envelopec(n) before calibration.

The MSE and SFDR before calibration are−17.80dB and25dB, and−36.51dB and43dB after calibration, respectively.Thus calibration yields about 18dB in MSE and SFDR im-provement.

Next, to assess the effect of the image band indexℓ on theperformance of the proposed calibration algorithm, we perform

8

−0.5 0 0.5−140

−120

−100

−80

−60

−40

−20

0

20

f

PSD

ofc(n)(dB)

Fig. 9. PSD of the estimated envelopec(n) after calibration.

simulations for values of the carrier frequencyFc equal to2.15, 3.15, 4.15 and 5.15GHz. The complex envelope signalcc(t) is given by (31) in all cases, and the sub-ADC samplingfrequency F ′

s = 1GHz, so that the values of the carrierfrequency correspond toℓ = 2, 3, 4 and 5, and ωb = 0.3πin all cases. For these signals, we select a nominal timingoffset value equal tod0 = 1/2 − 3/(8ℓ), which ensuresd0

does not correspond to a forbidden value. The gain and timingmismatches are stillγ = 10−2 and δ = −0.25 × 10−2. Forthese simulation parameters, the MSE of the complex envelopeerror before and after calibration and the final estimatesγ(L)and δ(L) are shown in Table I. The table indicates that theperformance of the complex envelope sampling scheme de-grades steadily both before and after calibration asℓ increases,and the MSE calibration improvement declines slightly from25dB for ℓ = 2 to 18dB for ℓ = 5. The gain underestimateincreases, but the accuracy of the the timing delay estimatedoes not suffer significantly whenℓ increases. Since othersimulations not shown here indicate that gain mismatches havelittle effect on the envelope MSE, we conclude that the mainreason for the loss of performance of the proposed calibrationalgorithm asℓ increases is not the blind estimation algorithm,which estimates timing mismatches accurately, but the first-order approximation (13) and (14) of the reconstruction filters,which becomes progressively less accurate asℓ increases. Thusfor large ℓ, it may be desirable to include second-order termsin expansions (13) and (14). The presence of large higher-order terms is also the cause of the underestimation of thegain mismatchγ, since the analysis presented in [9, Sec. 4]for gain estimation of baseband sampling TIADCs shows thatthe presence of noise contributes to a negative bias in the gainestimate. For the case considered here, the higher order terms

neglected in expansions (13) and (14) play the role of noise,which causes the gain mismatch underestimation.

Carrier Freq. 2.15GHz 3.15Gz 4.15GHz 5.15 GHzMSE, Before (dB) -25.51 -22.24 -19.76 -17.79MSE, After (dB) -50.89 -46.01 -40.97 -36.52γ(L)(×10−3) 10 9.7 8.8 7.7δ(L) (×10−3) 2.5 2.6 2.6 2.7

TABLE I. VARIATION OF THE MSE BEFORE AND AFTER CALIBRATION

AND OF THE MISMATCH PARAMETER ACCURACY AS THE IMAGE ZONE

INDEX ℓ INCREASES.

B. Bandlimited White Noise Compex Envelope

To demonstrate that the proposed blind estimation algorithmworks satisfactorily for different types of complex envelopesignals, for the case of a carrier frequencyFc = 5.15GHz,we consider the case where the complex envelopecc(t) isa bandlimited white noise signal with bandwidthB/(4π) =400MHz. The sub-ADC sampling frequency remainsF ′

s =1GHz, so that the oversampling ratio is againα = 0.2. ThePSD of the true sampled complex envelope is shown in Fig. 10.The nominal TIADC parametersd0 = 0.425 andg0 = 1, andmismatchesγ andδ are the same as before. Sinceωb = 0.3πand ℓ = 5, the reconstruction filtersHij(e

jω), i = 1, 2 andj = 0, 1 and bandpass filterHBP(ejω) are the same as in theearlier example.

−0.5 0 0.5−160

−140

−120

−100

−80

−60

−40

−20

f

PSD

ofc(n)(dB)

Fig. 10. PSD of the actual sampled bandlimited white noise complexenvelope.

The blind estimation algorithm is applied to a data blockof length L = 105 samples with adaptation step sizesµγ =5×10−4 andµδ = 5×10−6 and zero initial mismatch values.The resulting estimated mismatches are shown as a function

9

of time in Fig. 11 and Fig. 12. The final estimated gain andtiming offset mismatches estimates areγ(L) = 0.83 × 10−4

and δ(L) = −0.24 × 10−4. As in the case of a sum ofcomplex tones envelope, the gain mismatch is understimated,but slightly less so than in the complex tone case, and thetiming offset mismatch is estimated accurately. The MSE ofthe complex envelope error is−37.56dB.

0 1 2 3 4 5 6 7 8 9 10

x 104

0

1

2

3

4

5

6

7

8

9x 10

−3

Fig. 11. Gain mismatch estimates for a bandlimited white noise complexenvelope, withL = 105 samples, and adaptation step sizesµγ = 5× 10−4

amdµδ = 5× 10−6.

0 1 2 3 4 5 6 7 8 9 10

x 104

−2.5

−2

−1.5

−1

−0.5

0x 10

−3

Fig. 12. Timing offset mismatch estimates for a bandlimited whitenoisecomplex envelope, withL = 105 samples, and adaptation step sizesµγ =5× 10−4 amdµδ = 5× 10−6.

Finally, the PSD of the complex envelope after calibrationis shown in Fig. 13. The PSD is computed by selecting thefinal block of lengthN = 104 samples of the blind calibrationsimulation, after the estimation algorithm has converged.

−0.5 0 0.5−120

−110

−100

−90

−80

−70

−60

−50

−40

−30

−20

f

PSD

ofc(n)(dB)

Fig. 13. PSD of the sampled bandlimited white noise complex envelope aftercalibration.

V. CONCLUSIONS

A blind calibration method has been proposed for calibratinga nonuniformly interleaved ADC used for sampling directly thecomplex envelope of a bandpass signal. The blind calibrationmethod requires that the bandpass signal should be slightlyoversampled, which ensures the presence of a frequency bandover which the sampled complex envelopec(n) has no power.By extracting the component of the reconstructed envelopecorresponding to this frequency band, an error signal is ob-tained. This error signal can be minimized adaptively by usinga stochastic gradient approach in order to obtain estimatesofthe TIADC gain and timing offset mismatches.

The adaptive complex envelope reconstruction structureused in Fig. 4 relies onfixed FIR filters and includes onlythree adaptive taps (two equal toδ(n) and one equal toγ(n)).Numerical simulations for a complex envelope formed by sumsof complex tones, and for a bandlimited white noise complexenvelope show that the calibration scheme can be effectivefor low or moderate values of the image indexℓ. For largevalues values ofℓ, i.e. for large carrier frequencies, higherorder terms may be needed in the Taylor series approximations(13) and (14) of the reconstruction filters, which would in-crease significantly the complexity of the proposed calibrationmethod. Alternatively, one may consider estimating adaptivelythe entire FIR impulse responses of filtersH1 and andH2 tominimize errore(n). This approach has a high computationalcomplexity, but it could potentially correct additional sub-ADCimpairments, such as bandwidth mismatches [10], [15]–[17].

10

APPENDIX AIMPULSE RESPONSES OF THEGRADIENT FILTERS

The impulse responses of gradient filtersH11(ejω) and

H22(ejω) are obtained by taking the partial derivatives with

respect tod of the impulse responses of the correction filtersh1(n, d) andh2(n, d) given by (10) and (11). This gives

ℜh11(n, d) = ℜh21(n, d) = 0 ,

ℑh11(n, d) = π(2ℓ + sgn(ωb)) csc2(π(2ℓ + sgn(ωb))d)δ(n)

+π[2ℓ csc2(2πℓd) − (2ℓ + sgn(ωb)) csc2(π(2ℓ + sgn(ωb))d)]

×sin((π − |ωb|)n)

πn, (34)

and

ℑh21(n,θ) =1

gπ(n + d)

×[

(π − |ωb|)cos((π − |ωb|)(n + d))

sin(2πℓd)

−2πℓsin((π − |ωb|)(n + d)) cos(2πℓd)

sin2(2πℓd)

+|ωb|cos((π − |ωb|)(n + d) − πd)

sin(π(2ℓ + sgn(ωb))d)

+π(2ℓ + sgn(ωb)) sin((π − |ωb|)(n + d) − πd)

×cos(π(2ℓ + sgn(ωb))d)

sin2(π(2ℓ + sgn(ωb))d)

]

−1

n + dℑh2(n,θ) . (35)

APPENDIX BQUADRATIC APPROXIMATION OF J(θ)

To derive a quadratic approximation ofJ(θ), we start bynoting that the model of Fig. 2 implies that the DTFTsX1(e

jω)and X2(e

jω) of the sub-ADC sequences can be expressed interm of the DTFTC(ejω) of complex envelopec(n) as[

X1(ejω)

F−1(ejω)X2(ejω)

]

= M(ejω, d)

[

C(ej(ω−ωb))C∗(ej(ω+ωb))

]

(36)where

M(ejω, d) =1

2

[

1 1G(ejω) G∗(e−jω)

]

. (37)

Accordingly, the DTFT of signalr(n, θ) in Fig. 4 can beexpressed as

R(ejω, θ) =[

H1(ejω, d) H2(e

jω, θ)]

[

X1(ejω)

X2(ejω)

]

= L(ejω, θ)

[

C(ej(ω−ωb))C∗(ej(ω+ωb))

]

(38)

with

L(ejω, θ) = [ 1 0 ]M−1(ejω, d)

×

[

1 0

0 ggejω(d−d)

]

M(ejω, d) . (39)

By performing a first-order Taylor series expansion ofL(ejω, θ) in the vicinity of θ, we find

L(ejω, θ) ≈ [ 1 0 ]

+ǫLγ(ejω) + ηLδ(ejω) , (40)

withǫ = γ − γ , η = δ − δ ,

where

Lγ(ejω) = − [ 1 0 ]M−1(ejω, d)

×

[

0 00 1

]

M(ejω, d) (41)

and

Lδ(ejω) = [ 1 0 ]M−1(ejω, d)

×[

jω

[

0 00 1

]

M(ejω, d) −∂

∂dM(ejω, d)

]

. (42)

We find

∂

∂dM(ejω, d) =

1

2

[

0 0J(ejω)G(ejω) J∗(e−jω)G∗(e−jω)

]

(43)where

J(ejω)=

∂

∂dlnG(ejω)

is given by

J(ejω) =

−j2π(ℓ + 1) −π ≤ ω < −π + ωb

−j2πℓ −π + ωb ≤ ω < π ,

for ωb > 0 and

J(ejω) =

−j2πℓ −π ≤ ω ≤ π + ωb

−j2π(ℓ − 1) π + ωb ≤ ω < π .

for ωb < 0. Then by observing that the bandpass filterHBP(ejω) has been selected such that

HBP(ejω)C(ej(ω−ωb)) = 0 , (44)

we conclude that the DTFT of error signalE(ejω, θ) can beexpressed as

E(ejω, θ) = HBP(ejω)R(ejω, θ)

≈ [ǫNγ(ejω) + ηNδ(ejω)]C∗(e−j(ω+ωb)) , (45)

where after some substitutions, we find

Nγ(ejω) = HBP(ejω)Lγ(ejω)

[

01

]

= −HBP(ejω) [ 1 0 ]

×M−1(ejω, d)

[

0G∗(e−jω)/2

]

=1

2HBP(ejω)H1(e

jω, d) , (46)

11

and

Nδ(ejω) = HBP(ejω)Lδ(e

jω)

[

01

]

= (jω − J∗(e−jω))HBP(ejω) [ 1 0 ]

×M−1(ejω, d)

[

0G∗(e−jω)/2

]

=1

2(J∗(e−jω) − jω)HBP(ejω)H1(e

jω, d) . (47)

in expression (45),C∗(e−j(ω+ωb)) represents the Fouriertransform of signal

q(n) = e−jωbnc∗(n) .

To evaluate the power ofe(n, θ), we assume that the complexenvelope processc(n) is zero-mean WSS, i.e. that the complexautocorrelation function

Rc(m) = E[c(n + m)c∗(n)]

depends onm only. The power spectral density (PSD) ofc(n)is then defined as

Sc(ejω) =

∞∑

m=−∞

Rc(m)e−jmω (48)

It is 2π-periodic and since autocorrelationRc(m) satisfiesRc(m) = R∗

c(−m), it is real and non-negative (but notnecessarily even). Then the autocorrelation of signalq(n) is

Rq(n) = e−jωbmRc(−m)

and its PSD is

Rq(ejω) = Sc(e

−j(ω+ωb)) .

Expression (45) for the DTFT of error signale(n, θ) impliesthat it is also WSS with PSD

Se(n, θ) =[

ǫ2|Nγ(ejω)|2 + η2|Nδ(ejω)|2

]

Sq(ejω) ,

(49)where the absence of cross-term is due to the fact that the realpart of purely imaginary termJ∗(e−jω)−jω is zero. By usingthis expression inside (18), we find that

J(θ) =1

4π

∫ π

−π

Se(ejω, θ)dω

= Jγǫ2 + Jδη2 (50)

with

Jγ =1

4π

∫ π

−π

|Nγ(ejω)|2Sq(ejω)dω

=1

16π

∫

Im

|H1(ejω, d)|2Sc(e

−j(ω+ωb))dω (51)

and

Jδ =1

4π

∫ π

−π

|Nδ(ejω)|2Sq(e

jω)dω

=1

16π

∫

Im

|J∗(e−jω) − jω|2|H1(ejω, d)|2

×Sc(e−j(ω+ωb))dω . (52)

In equations (51) and (52),Im = I mod (2π) is defined sothatIm is a subset of[−π, π]. SinceH1(e

jω, d) is a piecewiseconstant and nonzero filter, the expressions (51) and (52) implythat the coefficientsJγ andJδ are nonzero, so thatJ(θ) is apositive definite quadratic function ofǫ and η in the vicinityof θ, as long as signalq(n) = ejωbnc∗(n) has some powerin bandIm. Equivalently,c(n) must have some power in theband

IT = [(1 − α)π, (1 + α)π] − 2ωb mod (2π) . (53)

In this caseJ(θ) will be minimized whenǫ = η = 0, orequivalently when estimatesγ = γ and δ = δ correspond tothe true mismatch values.

At this point, it is worth noting that the bandIT has width2απ and is completely contained inside the spectral supportregion [−(1−α)π, (1−α)π] of c(n) as long asαπ ≤ |ωb| ≤(1 − α)π. On the other hand whenωb is located either closeto the center of interval[−π, π] (with |ωb| < απ) or locatedclose to its edges (with|ωb ± π| < απ, only part of IT islocated within the spectral support region[−(1−α)π, (1−α)π]of c(n). When ωb is exactly equal to zero or±π, then IT

and the spectral support region[−(1 − α)π, (1 − α)π] aredisjoint, so that the calibration technique proposed here willnot work as described. To deal with such cases, it wouldbe useful to identify a frequency band other thanI wherer(n,θ) = e−jωbnc(n) is free of energy, and then in the blockdiagram of Fig. 4, change the passband of filterHBP to thiszero energy band, so that condition (44) still holds.

At this point it is worth indicating that although it has beenassumed thatc(n) is WSS in order to derive the closed-formexpressions (52) and (53) for the coefficientsJγ andJδ of thequadratic approximation (51) ofJ(θ), this assumption wasneeded only for analysis purposes. The conclusion thatJ(θ)is locally positive definite quadratic in the vicinity ofθ holdsin general, even ifc(n) is nonstationary, as long asc(n) hassignificant power in the frequency bandIT during segmentsof time which are long enough to ensure that the estimationalgorithm (22) converges.

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[1] B. C. Levy, M. S. Wahab, and A. Van Selow, “Direct complex envelopesampling with nonuniformly interleaved two-channel ADCs.” Submittedto IEEE TCAS II.

[2] A. Kohlenberg, “Exact interpolation of band-limited functions,” J.Applied Physics, vol. 24, pp. 1432–1436, Dec. 1953.

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[6] Y.-P. Lin and P. P. Vaidyanathan, “Periodically nonuniform sampling ofbandpass signals,”IEEE Trans. Sig. Proc., vol. 45, pp. 340–351, Mar.1998.

[7] W. Namgoong, “Finite-length synthesis filters for non-uniformly time-interleaved analog-to-digital converter,” inProc. 2002 IEEE Internat.Conf. Circuits and Systems, vol. 4, (Scottsdale, AZ), pp. 815–818, May2002.

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[18] F. Harris, X. Chen, E. Venosa, and F. A. N. Palmieri, “Two-channelTI-ADC for communication signals,” inProc. 2001 IEEE 12th Inter-nat. Workshop on Signal Proc. Advances in Wireless communications(SPAWC’ 11), (San Francisco, CA), pp. 576–580, June 2011.

[19] F. J. Harris, “Artifact-corrected time-interleaved ADC.” US Patent No.8,957,798, Feb. 2015.

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[21] H. L. Duc, C. Jabbour, P. Desgreys, O. Jamin, and V. T. Nguyen, “Afully digital background calibration of timing skew in undersamplingTI-ADC,” in Proc. 12th IEEE Internat. New Circuits and Systems Conf.(NEWCAS’14), (Trois Rivieres, Canada), pp. 53–56, June 2014.

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Bernard C. Levy received the diploma of IngenieurCivil des Mines from the Ecole Nationale Superieuredes Mines in Paris, France in 1974, and the Ph.D. inElectrical Engineering from Stanford University in1979.

From July 1979 to June 1987 he was Assistant andthen Associate Professor in the Department of Elec-trical Engineering and Computer Science at M.I.T.Since July 1987, he has been with the Universityof California at Davis, where he is Professor ofElectrical Engineering and a member of the Graduate

Group in Applied Mathematics. He served as Chair of the Department ofElectrical and Computer Engineering at UC Davis from 1996 to 2000. He wasa Visiting Scientist at the Institut de Recherche en Informatique et SystemesAl eatoires (IRISA) in Rennes, France from January to July 1993, and at theInstitut National de Recherche en Informatique et Automatique (INRIA), inRocquencourt, France, from September to December 2001. He is the author ofthe bookPrinciples of Signal Detection and Parameter Estimation(New York,NY: Springer, 2008). His research interests include statistical signal processing,estimation, detection, and circuits applications of signalprocessing.

Dr. Levy served as Associate Editor of the IEEE Transactionson Circuitsand Systems I, of the IEEE Transactions on Circuits and SystemsII, and of theEURASIP J. on Advances in Signal Processing. He is currentlyan AssociateEditor of Signal Processing. He is a member of SIAM.

Anthony Van Selow received his BS in ElectricalEngineering from UC Davis in 2012 and is currentlycompleting a MS in Electrical Engineering at UCDavis. He worked as an intern at Dysonics from 2012to 2014 and at Exponent from 2010 to 2011. Hisresearch interests include statistical signal process-ing, circuits applications of signal processing, andadaptive filtering.

Mansoor Wahab was born in Kabul, Afghanistan inJanuary 1993 and immigrated to the United States in2006. He completed his BS in Electrical Engineeringwith highest honors at UC Davis in June 2014.In his undergraduate studies he focused on signalprocessing and performed research on bandpass sam-pling and ADC calibration. He is currently a PHDstudent in the Electrical and Computer EngineeringDepartment at Cornell University where his researchhas shifted towards optimal control theory and itsapplications to multiagent systems. Mr. Wahab’s

honors include an ECE Department Citation from UC Davis and a CornellDean’s Excellence Fellowship for his first year of graduate studies. He is astudent member of IEEE and plans to pursue an academic career.